Mathematical Sciences: p-adic Automorphic Forms

这项研究涉及研究的p-adic对称空间，算术的模块化形式，以及一些计算和算法的问题，在数论和代数几何。在p-adic自守形式的一般领域中，将关于p-adic上半平面上的解析函数与测度及其边界上的测度之间关系的已知结果推广到了Drinfeld的高维p-adic上半空间.这样的结果有影响的几何代数簇是一致的德林费尔德空间，包括某些志村品种和德林费尔德模品种。它们也对一般线性群的p-adic表示理论产生了重要的影响。在模形式领域，首席研究员正在继续研究与“例外零猜想”有关的问题。最后，以前的工作算法计算皮卡德-富克斯微分方程和高斯-马宁连接的扩展。 本计画的研究范围为算术几何与自守形式的一般领域。代数几何是现代数学中最古老的部分之一。在过去的十年里，它已经发展到解决了几个世纪以来一直存在的问题。最初，它处理由最简单的方程，即多项式定义的平面中的图形。今天，该领域不仅利用代数方法，而且还利用分析和拓扑学方法;相反，它在这些领域中得到了广泛的应用。此外，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人技术等不同领域都很有用。自守形式产生于世纪中期的非欧几里德几何。因此，数学家和物理学家早就认识到，许多具有根本重要性的对象在其基本性质上是非欧几里德的。这一领域主要关注关于整数的问题，但在几何和分析的应用中，它保留了与其历史根源的联系，从而与理论物理中的规范理论和信息论中的编码理论等不同领域的问题保持联系。 这项研究涉及研究的p-adic对称空间，算术的模块化形式，以及一些计算和算法的问题，在数论和代数几何。在p-adic自守形式的一般领域中，将关于p-adic上半平面上的解析函数与测度及其边界上的测度之间关系的已知结果推广到了Drinfeld的高维p-adic上半空间.这样的结果讨厌的影响几何代数簇是一致的德林费尔德空间，包括某些志村品种和德林费尔德模品种。它们也对一般线性群的p-adic表示理论产生了重要的影响。在模形式领域，首席研究员正在继续研究与“例外零猜想”有关的问题。最后，以前的工作算法计算皮卡德-富克斯微分方程和高斯-马宁连接的扩展。 本计画的研究范围为算术几何与自守形式的一般领域。代数几何是现代数学中最古老的部分之一。在过去的十年里，它已经发展到解决了几个世纪以来一直存在的问题。最初，它处理由最简单的方程，即多项式定义的平面中的图形。今天，该领域不仅利用代数方法，而且还利用分析和拓扑学方法;相反，它在这些领域中得到了广泛的应用。此外，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人技术等不同领域都很有用。自守形式产生于世纪中期的非欧几里德几何。因此，B oth数学家和物理学家早就认识到，许多具有根本重要性的物体在其基本性质上是非欧几里德的。这一领域主要关注的是关于整数的问题，但在几何和分析的应用中，它保留了与其历史根源的联系，从而与理论物理中的规范理论和信息论中的编码理论等不同领域的问题保持联系。

This research involves the study of p-adic symmetric spaces, the arithmetic of modular forms, and a number of computational and algorithmic questions in number theory and algebraic geometry. In the general area of p-adic automorphic forms, known results about the relationship between analytic functions and measures on the p-adic upper half plane and measures on its boundary are generalized to Drinfeld's higher dimensional p-adic upper half spaces. Such results have implications for the geometry of algebraic varieties which are uniformized by the Drinfeld spaces, including certain Shimura varieties and Drinfeld modular varieties. They also have important consequences for the p-adic representation theory of the general linear group. In the area of modular forms, the principal investigator is continuing to study questions relating to the "exceptional zero conjecture". Finally, earlier work on algorithms for computing Picard-Fuchs differential equations and the Gauss-Manin connection are extended. The research in this project lies in the general areas of arithmetic geometry and automorphic forms. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; conversely, it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. Automorphic forms arose out of non-Euclidean geometry in the middle of the nineteenth century. Both mathematicians and physicists have thus long realized that many objects of fundamental importance are non-Euclidean in their basic nature. This field is principally concerned with questions about the whole numbe rs, but in its use of geometry and analysis, it retains connection to its historical roots and thus to problems in areas as diverse as gauge theory in theoretical physics and coding theory in information theory. This research involves the study of p-adic symmetric spaces, the arithmetic of modular forms, and a number of computational and algorithmic questions in number theory and algebraic geometry. In the general area of p-adic automorphic forms, known results about the relationship between analytic functions and measures on the p-adic upper half plane and measures on its boundary are generalized to Drinfeld's higher dimensional p-adic upper half spaces. Such results hate implications for the geometry of algebraic varieties which are uniformized by the Drinfeld spaces, including certain Shimura varieties and Drinfeld modular varieties. They also have important consequences for the p-adic representation theory of the general linear group. In the area of modular forms, the principal investigator is continuing to study questions relating to the "exceptional zero conjecture". Finally, earlier work on algorithms for computing Picard-Fuchs differential equations and the Gauss-Manin connection are extended. The research in this project lies in the general areas of arithmetic geometry and automorphic forms. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; conversely, it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. Automorphic forms arose out of non-Euclidean geometry in the middle of the nineteenth century. B oth mathematicians and physicists have thus long realized that many objects of fundamental importance are non-Euclidean in their basic nature. This field is principally concerned with questions about the whole numbers, but in its use of geometry and analysis, it retains connection to its historical roots and thus to problems in areas as diverse as gauge theory in theoretical physics and coding theory in information theory.